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We analyze the perturbations T + B of a selfadjoint operator T in a Hilbert space H with discrete spectrum \({\{ t_k\}, T \phi_k = t_k \phi_k}\). In particular, if t  k+1 − t  k  ≥ ck  α - 1, α > 1/2 and \({\| B \phi_k \| = o(k^{\alpha - 1})}\) then the system of root vectors of T + B, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in H (Theorem 6). Under the assumptions \({t_{k+p} - t_k \geq d  > 0, \forall k}\) (with d and p fixed) and \({\| B \phi_k \| \rightarrow 0}\) a Riesz system {P  k } of projections on invariant subspaces of T + B, Rank P  k  ≤ p, is constructed (Theorem 3).

Root System of a Perturbation of a Selfadjoint Operator with Discrete Spectrum